Let's begin by analyzing the given information and the desired outcome of our proof. We want to show that △ PRQ is congruent to △ TRS. Recall that by the definition of congruent polygons, we want to show that the sides and the angles of these triangles are congruent.
Congruent sides
Congruent angles
PQ ≅ TS
∠ P ≅ ∠ T
PR ≅ TR
∠ Q ≅ ∠ S
RQ ≅ RS
∠ PRQ ≅ ∠ TRS
We are given that TR ≅ PR and ∠ S ≅ ∠ Q. By the Symmetric Property of Congruence, we know that PR ≅ TR and ∠ Q ≅ ∠ S. We are also given that PQ ≅ TS and ∠ P ≅ ∠ T. Therefore, we need to prove that RQ ≅ RS and ∠ PRQ ≅ ∠ TRS. Let's begin with showing that all angles are congruent.
Statement 1)& ∠ P ≅ ∠ T and ∠ Q ≅ ∠ S Reason 1)& Given
Notice that ∠ PRQ and ∠ TRS are vertical angles. By the Vertical Angles Theorem, we can conclude that they are congruent.
Statement 2)& ∠ PRQ ≅ ∠ TRS Reason 2)& Vertical Angles Theorem
Next, we will prove that RQ ≅ RS. First, notice that we can rewrite PR ≅ TR as RP ≅ RT. This is because PR and RP are both names for one side that has the endpoints P and R. We will continue our proof by stating the next given information, RT ≅ RS.
Statement 3)& RT ≅ RS Reason 3)& Given
Now, let's summarize what we know about the congruent sides so far.
RP ≅ RT and RT ≅ RS
By the Transitive Property of Congruence, we can conclude that RP ≅ RS.
Statement 4)& RP ≅ RS Reason 4)& Transitive Prop. of Congruence
We are also given that RP ≅ RQ. By the Symmetric Property of Congruence, it means that RQ ≅ RP.
Statement 5)& RQ ≅ RP Reason 5)& Given
Let's summarize what we know from the two last statements.
RQ ≅ RP and RP ≅ RS
Again, by the Transitive Property of Congruence we can conclude that RQ ≅ RS.
Statement 6)& RQ ≅ RS Reason 6)& Transitive Prop. of Congruence
We have shown that all the sides and the angles in the triangles are congruent!
Congruent sides
Congruent angles
PQ ≅ TS
∠ P ≅ ∠ T
PR ≅ TR
∠ Q ≅ ∠ S
RQ ≅ RS
∠ PRQ ≅ ∠ TRS
Therefore, by the definition of congruent polygons △ PRQ ≅ △ TRS.
Statement 7)& △ PRQ ≅ △ TRS Reason 7)& Definition of congruent polygons
Finally, we can complete our two-column table!
